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A COMPACTIFICATION OVER Mg of the UNIVERSAL MODULI SPACE of SLOPE-SEMISTABLE VECTOR BUNDLES

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A COMPACTIFICATION OVER Mg of the UNIVERSAL MODULI SPACE of SLOPE-SEMISTABLE VECTOR BUNDLES JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 9, Number 2, April 1996 A COMPACTIFICATION OVER Mg OF THE UNIVERSAL MODULI SPACE OF SLOPE-SEMISTABLE VECTOR BUNDLES RAHUL PANDHARIPANDE Contents 0. Introduction 425 1. The quotient construction 428 2. The fiberwise G.I.T. problem 433 3. Cohomology bounds 435 4. Slope-unstable, torsion free sheaves 437 5. Special, torsion bounded sheaves 440 6. Slope-semistable, torsion free sheaves 443 7. Two results in geometric invariant theory 449 8. The construction of Ug(e, r) 454 9. Basic properties of Ug(e, r) 458 10. The isomorphism between Ug(e, 1) and Pg,e 464 References 470 0. Introduction 0.1. Compactifications of moduli problems. Initial statements of moduli prob- lems in algebraic geometry often do not yield compact moduli spaces. For example, the moduli space Mg of nonsingular, genus g 2 curves is open. Compact moduli spaces are desired for several reasons. Degeneration≥ arguments in moduli require compact spaces. Also, there are more techniques available to study the global ge- ometry of compact spaces. It is therefore valuable to find natural compactifications of open moduli problems. In the case of Mg, there is a remarkable compactification due to P. Deligne and D. Mumford. A connected, reduced, nodal curve C of arith- metic genus g 2isDeligne-Mumford stable if each nonsingular rational component ≥ contains at least three nodes of C. Mg, the moduli space of Deligne-Mumford stable genus g curves, is compact and includes Mg as a dense open set. There is a natural notion of stability for a vector bundle E on a nonsingular curve C.Lettheslopeµbe defined as follows: µ(E)=degree(E)/rank(E). E is slope-stable (slope-semistable) if (1) µ(F ) < ( ) µ(E) ≤ Received by the editors April 18, 1994 and, in revised form, December 8, 1994. 1991 Mathematics Subject Classification. Primary 14D20, 14J10. Supported by an NSF Graduate Fellowship. c 1996 American Mathematical Society 425 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 426 RAHUL PANDHARIPANDE for every proper subbundle F of E. When the degree and rank are not coprime, the moduli space of slope-stable bundles is open. UC(e, r), the moduli space of slope- semistable bundles of degree e and rank r,iscompact.AnopensetofUC(e, r) corresponds bijectively to isomorphism classes of stable bundles. In general, points of UC(e, r) correspond to equivalence classes (see section (1.1) ) of semistable bun- dles. The moduli problem of pairs (C, E), where E is a slope-semistable vector bundle on a nonsingular curve C, cannot be compact. No allowance is made for curves that degenerate to nodal curves. A natural compactification of this moduli problem of pairs is presented here. 0.2. Compactification of the moduli problem of pairs. let C be a fixed algebraically closed field. As before, let Mg be the moduli space of nonsingular, complete, irreducible, genus g 2 curves over the field C. For each [C] Mg, ≥ ∈ there is a natural projective variety, UC(e, r), parametrizing degree e,rankr,slope- semistable vector bundles (up to equivalence) on C.Forg 2, let Ug(e, r)betheset ≥ of equivalence classes of pairs (C, E), where [C] Mg and E is a slope-semistable vector bundle on C of the specified degree and rank.∈ A good compactification, K, of the moduli set of pairs Ug(e, r) should satisfy at least the following conditions: (i) K is a projective variety that functorially parametrizes a class of geometric objects. (ii) Ug(e, r) functorially corresponds to an open dense subset of K. (iii) There exists a morphism η : K Mg such that the natural diagram com- mutes: → Ug(e, r) K −−−−→ η M g M g y −−−−→ y (iv) For each [C] Mg, there exists a functorial isomorphism ∈ 1 η− ([C]) ∼= UC (e, r)/Aut(C). The main result of this paper is the construction of a projective variety Ug(e, r) that parametrizes equivalence classes of slope-semistable, torsion free sheaves on Deligne-Mumford stable, genus g curves and satisfies conditions (i-iv) above. The definition of slope-semistability of torsion free sheaves (due to C. Seshadri) is given in section (1.1). 0.3. The method of construction. An often successful approach to moduli con- structions in algebraic geometry involves two steps. In the first step, extra data is added to rigidify the moduli problem. With the additional data, the new moduli problem is solved by a Hilbert or Quot scheme. In the second step, the extra data is removed by a group quotient. Geometric Invariant Theory is used to study the quotient problem in the category of algebraic schemes. In good cases, the final quotient is the desired moduli space. In order to rigidify the moduli problem of pairs, the following data is added to (C, E): (i) An isomorphism CN+1 ∼ H0(C, ω10), → C (ii) An isomorphism Cn ∼ H0(C, E). → License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A COMPACTIFICATION OVER Mg OF THE UNIVERSAL MODULI SPACE 427 Note ωC is the canonical bundle of C. The numerical invariants of the moduli problem of pairs are the genus g, degree e, and rank r. The rigidified problem should have no more numerical invariants. Hence, N and n must be determined by g, e,andr. Certainly, N = 10(2g 2) g by Riemann-Roch. It is assumed H1(E)=0andEis generated by global− sections.− In the end, it is checked these assumptions are consequences of the stability condition for sufficiently high degree bundles. We see n = χ(E)=e+r(1 g). − N N The isomorphism of (i) canonically embeds C in P = PC . The isomorphism of (ii) exhibits E as a canonical quotient n C C E 0. ⊗O → → The basic parameter spaces in algebraic geometry are Hilbert and Quot schemes. Subschemes of a fixed scheme X are parametrized by Hilbert schemes Hilb(X). Quotients of a fixed sheaf F on X are parametrized by Quot schemes Quot(X, F). The rigidified curve C can be parametrized by a Hilbert scheme H of curves in PN , and the rigidified bundle E can be parametrized by a Quot scheme n Quot(C, C C) of quotients on C. In fact, Quot schemes can be defined in ⊗O a relative context. Let UH be the universal curve over the Hilbert scheme H.The n N family of Quot schemes, Quot(C, C C), defined as C, P varies in H is simply the relative Quot scheme of the⊗O universal curve over→ the Hilbert scheme: n Quot(UH H, C UH). This relative Quot scheme is the parameter space of the rigidified→ pairs (up⊗O to scalars). The Quot scheme set up is discussed in detail in section (1.3). N+1 n The actions of GLN+1(C)onC and GLn(C)onC yieldanactionof GLN+1 GLn on the rigidified data. There is an induced product action on the relative× Quot scheme. It is easily seen the scalar elements of the groups act trivially on the Quot scheme. Ug(e, r) is constructed via the quotient: n (2) Quot(UH H, C U )/SLN+1 SLn. → ⊗O H × There is a projection of the rigidified problem of pairs (C, E) with isomorphisms (i) and (ii) to a rigidified moduli problem of curves {C with isomorphism (i) . } { } The projection is GLN+1-equivariant with respect to the natural GLN+1-action on the rigidified data of curves. The Hilbert scheme H is a parameter space of the rigidified problem of curves (up to scalars). By results of Gieseker ([Gi]) reviewed in section (1.2), the quotient H/SLN+1 is Mg. A natural morphism Ug(e, r) Mg is therefore obtained. Gieseker’s results require the choice in isomorphism (i)→ of at least the 10-canonical series. The technical heart of the paper is the study of the Geometric Invariant Theory problem (2). The method is divide and conquer. The action of SLn alone is first studied. The SLn-action is called the fiberwise G.I.T. problem. It is solved in sections (2) - (6). The action of SLN+1 alone is then considered. There are two pieces in the study of the SLN+1 action. First, Gieseker’s results in [Gi] are used in an essential way. Second, the abstract G.I.T. problem of SLN+1 acting on P(Z) × P(W )whereZ,Ware representations of SLN+1 is studied. If the linearization is taken to be P(Z)(k) P(W)(1) where k>>1, there are elementary set theoretic relationshipsO between⊗O the stable and unstable loci of P(Z)andP(Z) P(W). These relationships are determined in section (7). Roughly speaking, the abstract× lemmas are used to import the invariants Gieseker has determined in [Gi] to the problem at License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 428 RAHUL PANDHARIPANDE hand. In section (8), the solution of fiberwise problem is combined with the study of the SLN+1-action to solve the product G.I.T. problem (2). 0.4. Relationship with past results. In [G-M], D. Gieseker and I. Morrison propose a different approach to a compactification over Mg of the universal moduli space of slope-semistable bundles. The moduli problem of pairs is rigidified by adding only the data of an isomorphism Cn H0(C, E). By further assumptions on E, an embedding into a Grassmannian is→ obtained 0 C, G(rank(E),H (C, E)∗).
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